Synchronization of networked, or coupled, systems has been examined for large networks of identical (Pikovsky, A and Rosenblum, M and Kurths, J, “Synchronization: A universal concept in nonlinear science,” Cambridge university press, Cambridge, 2001) and heterogeneous oscillators, for example as described in Restrepo, J G and Ott, E and Hunt, B R, “Emergence of coherence in complex networks of heterogeneous dynamical systems”, PHYSICAL REVIEW LETTERS, Volume 96, Number 25, Article-Number 254103 (Jun. 30, 2006). For coupled systems with smaller numbers of oscillators, several new dynamical phenomena have been observed, including generalized (Rulkov, N F and Sushchik, M M and Tsimring, L S and Abarbanel, H D I, “Generalized synchronization of chaos in directionally coupled chaotic systems”, PHYSICAL REVIEW E, Volume 51, Number 2, Pages 980-994 (February 1995)), phase (Rosenblum, M G and Pikovsky, A S and Kurths, J, “Phase synchronization of chaotic oscillators”, PHYSICAL REVIEW LETTERS, Volume 76, Number 11, Pages 1804-1807 (Mar. 11, 1996)), and lag (Rosenblum, M G and Pikovsky, A S and Kurths, J, “From phase to lag synchronization in coupled chaotic oscillators”, PHYSICAL REVIEW LETTERS, Volume 78, Number 22, Pages 4193-4196 (Jun. 2, 1997)) (hereinafter “Rosenblum et al.”) synchronization. Lag synchronization, in which there is a phase shift between observed signals, is one of the routes to complete synchrony as coupling is increased (Rosenblum et al.) and may occur without the presence of delay in the coupling terms.
For systems with delayed coupling, a time lag between the oscillators is typically observed, with a leading time series followed by a lagging one. Such lagged systems are said to exhibit achronal synchronization. In Heil, T and Fischer, I and Elsasser, W and Mulet, J and Mirasso, C R, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers”, PHYSICAL REVIEW LETTERS, Volume 86, Number 5, Pages 795-798 (Jan. 29, 2001), the existence of achronal synchronization in a mutually delay-coupled semiconductor laser system was shown experimentally, and in White, J K and Matus, M and Moloney, J V, “Achronal generalized synchronization in mutually coupled semiconductor lasers”, PHYSICAL REVIEW E, Volume 65, Number 3, Part 2A, Article-Number 036229 (March 2002), studied theoretically in a single-mode semiconductor laser model. In the case of short coupling delay for unidirectionally coupled systems, anticipatory synchronization occurs when a response in a system's state is not replicated simultaneously but instead is anticipated by the response system (Voss, H U, “Anticipating chaotic synchronization”, PHYSICAL REVIEW E, Volume 61, Number 5, Part A, Pages 5115-5119 (May 2000); and Voss, Hu, “Dynamic long-term anticipation of chaotic states”, PHYSICAL REVIEW LETTERS, Volume 8701, Number 1, Article-Number 014102 (Jul. 2, 2001)), and an example of anticipation in synchronization is found in coupled semiconductor lasers (Masoller, C, “Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback”, PHYSICAL REVIEW LETTERS, Volume 86, Number 13, Pages 2782-2785 (Mar. 26, 2001)). Cross-correlation statistics between the two intensities showed clear maxima at delay times consisting of the difference between the feedback and the coupling delay. Anticipatory responses in the presence of stochastic effects have been observed in models of excitable media (Ciszak, M and Calvo, O and Masoller, C and Mirasso, C R and Toral, R, “Anticipating the response of excitable systems driven by random forcing”, PHYSICAL REVIEW LETTERS, Volume 90, Number 20, Article-Number 204102 (May 23, 2003)). Noise further complicates the picture in that theory and experiment may exhibit achronal synchronization, with switching between leader and follower (Mulet, J and Mirasso, C and Heil, T and Fischer, I, “Synchronization scenario of two distant mutually coupled semiconductor lasers”, JOURNAL OF OPTICS B-QUANTUM AND SEMICLASSICAL OPTICS, Volume 6, Number 1, Pages 97-105 (January 2004)).
Given that both lag and anticipatory dynamics may be observed in delay-coupled systems, it is natural to ask whether the isochronal (zero-lag) state, in which there is no phase difference in the synchronized time series, may be stabilized in coupled systems. A recent example of stable isochronal synchronization may be found in Klein, Einat and Gross, Noam and Rosenbluh, Michael and Kinzel, Wolfgang and Khaykovich, Lev and Kanter, Ido}, “Stable isochronal synchronization of mutually coupled chaotic lasers”, PHYSICAL REVIEW E, Volume 73, Number 6, Part 2, Article-Number 066214 (June 2006), incorporated herein by reference, which considers two coherently coupled semiconductor lasers.